Finite Group Databasebeta

← #22562979

Subgroups — #22562979

Order 8576.

476 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14524036C211yesmaximal, characteristic
2#16863291C211yesmaximal, characteristic, complemented
3#16863291C211yesmaximal, characteristic, complemented
4#16863289C211yesmaximal, characteristic
5#16863284C211yesmaximal, characteristic
6#16863284C211yesmaximal, characteristic
7#16863283C211yesmaximal, characteristic, complemented
8#16863279C211yesmaximal, characteristic, complemented
9#16863278C211yesmaximal, characteristic, complemented
10#16863269C211yesmaximal, characteristic
11#16863261C211yesmaximal, characteristic, complemented
12C4⋊C4×Dic67C211yesmaximal, characteristic, complemented
13#16863247C211yesmaximal, characteristic, complemented
14C4×C22⋊Dic67C211yesmaximal, characteristic
15#385×C67C211yesmaximal, characteristic
16#14524036C418yescomplemented
17C2×C4⋊Dic67C2211yescharacteristic
18C2×C4⋊Dic67C2211yescharacteristic
19C2×Dic67⋊C4C2211yescharacteristic
20C2×Dic67⋊C4C2211yescharacteristic
21C2×Dic67⋊C4C2211yescharacteristic
22C2×Dic67⋊C4C2211yescharacteristic
23C2×Dic67⋊C4C2211yescharacteristic
24C2×Dic67⋊C4C2211yescharacteristic
25C2×C22⋊Dic67C2211yescharacteristic
26C2×C22⋊Dic67C2211yescharacteristic
27C2×C22⋊Dic67C2211yescharacteristic
28C2×C22⋊Dic67C2211yescharacteristic
29C2×C4×Dic67C2211yescharacteristic
30C2×C4×Dic67C2211yescharacteristic
31C2×C4×Dic67C2211yescharacteristic
32C2×C4×Dic67C2211yescharacteristic
33C2×C4×Dic67C2211yescharacteristic
34C2×C4×Dic67C2211yescharacteristic
35C67⋊#95C2211yescharacteristic
36C67⋊#95C2211yescharacteristic
37C67⋊#95C2211yescharacteristic
38C67⋊#95C2211yescharacteristic
39C67⋊#95C2211yescharacteristic
40C67⋊#95C2211yescharacteristic
41C67⋊#95C2211yescharacteristic
42C67⋊#95C2211yescharacteristic
43C67⋊#95C2211yescharacteristic
44C67⋊#95C2211yescharacteristic
45C23×C4×C67C2211yescharacteristic
46C2×C4⋊C4×C67C2211yescharacteristic
47C2×C4⋊C4×C67C2211yescharacteristic
48C2×C22⋊C4×C67C2211yescharacteristic
49C2×C22⋊C4×C67C2211yescharacteristic
50#95×C67C2211yescharacteristic
51#95×C67C2211yescharacteristic
52C22×Dic67C2311yescharacteristic
53C22×Dic67C2311yescharacteristic
54C22×Dic67C2311yescharacteristic
55C22×Dic67C2311yescharacteristic
56C22×Dic67C2311yescharacteristic
57C22×Dic67C2311yescharacteristic
58C22×Dic67C2311yescharacteristic
59C22×Dic67C2311yescharacteristic
60C4⋊Dic67—41no—
61C4⋊Dic67C2×C414yescomplemented
62Dic67⋊C4—22no—
63Dic67⋊C4—22no—
64Dic67⋊C4—41no—
65Dic67⋊C4—41no—
66Dic67⋊C4C2×C414yescomplemented
67Dic67⋊C4C2×C414yescomplemented
68C22⋊Dic67—22no—
69C22⋊Dic67—41no—
70C22⋊Dic67C2×C414yes—
71C22⋊Dic67C2×C414yes—
72C4×Dic67—22no—
73C4×Dic67—41no—
74C4×Dic67—41no—
75C4×Dic67—41no—
76C4×Dic67—41no—
77C4×Dic67C2×C414yescomplemented
78C24×C67C2311yescharacteristic
79C22×C4×C67—22no—
80C22×C4×C67—41no—
81C22×C4×C67—41no—
82C22×C4×C67C2311yescharacteristic
83C22×C4×C67C2311yescharacteristic
84C22×C4×C67C2311yescharacteristic
85C22×C4×C67C2311yescharacteristic
86C22×C4×C67C2311yescharacteristic
87C22×C4×C67C2311yescharacteristic
88C4⋊C4×C67—22no—
89C4⋊C4×C67—41no—
90C22⋊C4×C67—41no—
91C22⋊C4×C67C2×C414yes—
92C23×C67—21no—
93C23×C67—21no—
94C23×C67—21no—
95C23×C67—21no—
96C23×C67—21no—
97C23×C67—21no—
98C23×C67C22×C412yes—
99C23×C67C2411yescharacteristic
100C2×C4×C67—21no—
101C2×C4×C67—21no—
102C2×C4×C67—21no—
103C2×C4×C67—21no—
104C2×C4×C67—21no—
105C2×C4×C67—21no—
106C2×C4×C67—21no—
107C2×C4×C67—21no—
108C2×C4×C67—21no—
109C2×C4×C67—21no—
110C2×C4×C67—21no—
111C2×C4×C67—21no—
112C2×C4×C67—21no—
113C2×C4×C67—21no—
114C2×C4×C67—21no—
115C2×C4×C67—44no—
116C2×C4×C67C22×C412yes—
117C2×C4×C67C22×C412yes—
118C2×C4×C67#4112yes—
119C2×Dic67—21no—
120C2×Dic67—21no—
121C2×Dic67—21no—
122C2×Dic67—21no—
123C2×Dic67—21no—
124C2×Dic67—21no—
125C2×Dic67—21no—
126C2×Dic67—21no—
127C2×Dic67—21no—
128C2×Dic67—21no—
129C2×Dic67—21no—
130C2×Dic67—21no—
131C2×Dic67—21no—
132C2×Dic67—21no—
133C2×Dic67—21no—
134C2×Dic67—21no—
135C2×Dic67—21no—
136C2×Dic67—21no—
137C2×Dic67—21no—
138C2×Dic67C22×C412yes—
139C2×Dic67C22×C412yes—
140C2×Dic67C22×C412yes—
141C2×Dic67C22×C412yes—
142C2×Dic67#4112yes—
143C22×C67—22no—
144C22×C67—22no—
145C22×C67—22no—
146C22×C67—41no—
147C22×C67—41no—
148C22×C67—41no—
149C22×C67—41no—
150C22×C67C23×C411yescharacteristic
151C22×C67C2×#4111yescharacteristic
152C22×C67C2×#4111yescharacteristic
153C22×C67H(5,𝔽2)11yescharacteristic
154C22×C67H(5,𝔽2)11yescharacteristic
155C22×C672−1+411yescharacteristic
156C22×C672−1+411yescharacteristic
157C4×C67—22no—
158C4×C67—22no—
159C4×C67—22no—
160C4×C67—41no—
161C4×C67—41no—
162C4×C67—41no—
163Dic67—22no—
164Dic67—22no—
165Dic67—22no—
166Dic67—22no—
167Dic67—22no—
168Dic67—41no—
169Dic67—41no—
170Dic67—41no—
171C2×C67—42no—
172C2×C67C4×#4111yescharacteristic
173C2×C67#52011yescharacteristic
174C2×C67#52011yescharacteristic
175C2×C67#53811yescharacteristic
176C2×C67#53911yescharacteristic
177C2×C67#54111yescharacteristic
178C2×C67#54211yescharacteristic
179#2334—671nomaximal, Sylow
180C67#233411yescharacteristic, complemented, Sylow
181C2×#126—671no—
182#389—671no—
183#389—671no—
184#388—671no—
185#388—671no—
186#388—671no—
187#387—671no—
188#387—671no—
189#387—671no—
190#385—671no—
191#385D6711yescharacteristic
192#382—671no—
193C4×C4⋊C4—671no—
194C4×C4⋊C4—671no—
195C4×C22⋊C4—671no—
196C23×C4C2×D6711yescharacteristic
197#126—678no—
198C2×C4⋊C4—671no—
199C2×C4⋊C4—671no—
200C2×C4⋊C4—671no—
201C2×C4⋊C4—671no—
202C2×C4⋊C4—671no—
203C2×C4⋊C4—671no—
204C2×C4⋊C4—671no—
205C2×C4⋊C4—671no—
206C2×C4⋊C4C2×D6711yescharacteristic
207C2×C4⋊C4C2×D6711yescharacteristic
208C2×C22⋊C4—671no—
209C2×C22⋊C4—671no—
210C2×C22⋊C4—671no—
211C2×C22⋊C4—671no—
212C2×C22⋊C4C2×D6711yescharacteristic
213C2×C22⋊C4C2×D6711yescharacteristic
214C2×C42—671no—
215C2×C42—671no—
216C2×C42—671no—
217C2×C42—671no—
218C2×C42—671no—
219C2×C42—671no—
220#95—671no—
221#95—671no—
222#95—671no—
223#95—671no—
224#95—671no—
225#95—671no—
226#95—671no—
227#95—671no—
228#95—671no—
229#95—671no—
230#95C2×D6711yescharacteristic
231#95C2×D6711yescharacteristic
232C24C22×D6711yescharacteristic
233C22×C4—22no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—671no—
237C22×C4—671no—
238C22×C4—671no—
239C22×C4—671no—
240C22×C4—671no—
241C22×C4—671no—
242C22×C4—671no—
243C22×C4—671no—
244C22×C4C22×D6711yescharacteristic
245C22×C4C22×D6711yescharacteristic
246C22×C4C22×D6711yescharacteristic
247C22×C4C22×D6711yescharacteristic
248C22×C4C22×D6711yescharacteristic
249C22×C4C22×D6711yescharacteristic
250C4⋊C4—22no—
251C4⋊C4—41no—
252C4⋊C4—2681no—
253C4⋊C4—2681no—
254C4⋊C4—2681no—
255C4⋊C4—674no—
256C4⋊C4—674no—
257C4⋊C4—674no—
258C4⋊C4—1342no—
259C4⋊C4—1342no—
260C22⋊C4—41no—
261C22⋊C4—2681no—
262C22⋊C4—674no—
263C22⋊C4—674no—
264C22⋊C4—1342no—
265C22⋊C4C4×D6714yes—
266C42—2681no—
267C42—2681no—
268C42—2681no—
269C42—2681no—
270C42—674no—
271C42—1342no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23C2×C4×D6712yes—
279C23C23×D6711yescharacteristic
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—44no—
296C2×C4—672no—
297C2×C4—672no—
298C2×C4—672no—
299C2×C4—672no—
300C2×C4—672no—
301C2×C4—1341no—
302C2×C4—1341no—
303C2×C4—1341no—
304C2×C4—1341no—
305C2×C4—1341no—
306C2×C4—1341no—
307C2×C4—1341no—
308C2×C4—1341no—
309C2×C4—1341no—
310C2×C4—1341no—
311C2×C4—1341no—
312C2×C4—1341no—
313C2×C4—1341no—
314C2×C4—1341no—
315C2×C4—1341no—
316C2×C4—1341no—
317C2×C4—1341no—
318C2×C4—1341no—
319C2×C4—1341no—
320C2×C4C2×C4×D6712yes—
321C2×C4C2×C4×D6712yes—
322C2×C4#1176162712yes—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22C22×C4×D6711yescharacteristic
331C22#41×D6711yescharacteristic
332C22C2×#1176162711yescharacteristic
333C22#1452408811yescharacteristic
334C22#1452408811yescharacteristic
335C22#1452409011yescharacteristic
336C22#1452409111yescharacteristic
337C4—22no—
338C4—22no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—2681no—
344C4—2681no—
345C4—2681no—
346C4—1342no—
347C4—1342no—
348C4—1342no—
349C4—1342no—
350C4—1342no—
351C2—42no—
352C2C4×#1176162711yescharacteristic
353C2#1686387511yescharacteristic
354C2#1686387711yescharacteristic
355C2#1686398111yescharacteristic
356C2#1686398911yescharacteristic
357C2#1686400611yescharacteristic
358C2#1686401611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D67
commutatorC22×C67C23×C4
Fitting#385×C67C2
socleC23×C67C24
hypercenter#385D67
hypocenterC67#2334
LCSG ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ #385

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #16863291C2semidirect
3. #16863291C2semidirect
7. #16863283C2semidirect
8. #16863279C2semidirect
9. #16863278C2semidirect
11. #16863261C2semidirect
12. C4⋊C4×Dic67C2semidirect
13. #16863247C2semidirect
16. #14524036C4semidirect
61. C4⋊Dic67C2×C4semidirect
66. Dic67⋊C4C2×C4semidirect
67. Dic67⋊C4C2×C4semidirect
77. C4×Dic67C2×C4semidirect
180. C67#2334semidirect